The Zak Phase is Half the Signature of the Spectral Localizer

Abstract

The topological index, or Zak phase, of an insulating system is an important physical invariant, related to its Hall Conductance and topological classification (See: Section 3 of [Sha24b], as well as [Zak89]). In a series of papers, Terry Loring and Hermann Schulz-Baldes introduced a method to calculate the topological index of a gapped insulator as the half-signature of a finite-dimensional matrix, known as the "spectral localizer" (See: [LS17],[LS19], [VSS19], [LS20]). They proffered two main proof schemes for their method: in [LS17] and [LS20], they approached the problem K-theoretically, while in [LS19] and [VSS19] they leveraged the theory of spectral flow. Both techniques require the insulator to be gapped, while numerics show that the spectral localizer method should still work in the mobility gap regime, with Anderson Localization (See: [VSS19]). On the other hand, the functional-analytic nature of the objects involved indicates that a simpler, more instructive approach should be available. In this work, we shall outline a new approach to the spectral localizer, demonstrate its validity numerically, and show its proof for a subset of periodic systems. We will also report on the prospects for future work extending the results herein, with the hope of eventually attacking the case of the mobility gap regime, where the tools of K-Theory are inapplicable. The main principle underlying our new framework is the Bulk-Edge Correspondence, as it applies to a two-dimensional system determined by the spectral localizer.